A102643 -id:A102643 - cp's OEIS frontend

A102644 A006530(x)=2 is a local minimum if x=2^n. Running downward with argument x started at 2^n, the largest prime divisor should increase. The value of first peak is a(n).

2, 3, 7, 13, 31, 61, 127, 127, 73, 1021, 89, 4093, 8191, 16381, 151, 257, 131071, 131071, 524287, 1048573, 337, 683, 178481, 16777213, 1801, 8191, 262657, 1877171, 2089, 46684427, 2147483647, 2147483647, 599479, 3360037, 6871947673, 283007

Offset: 1

Labos Elemer, Jan 21 2005

nonn

We may call these terms "downward-zenith-primes" belonging to 2^n-s. They do not exceed previous-primes before 2^n [A014234(n)].

			n=20: 2^20=1048576; the largest prime divisors for arguments if running downward from 2^20 are as follows: {2,41,524287,1048573,73}.
The first lower peak before argument 2^20=1048576 is a(20)=1048573.
n=1: a(1)=2 the peak equals the central value because there are no prime divisors>0 below n=2^1=2.

Michael De Vlieger, Table of n, a(n) for n = 1..150

Cf. A006530, A102640, A102641, A102642, A102643, A014234.

Table[2 + Total@ TakeWhile[Differences@ Map[FactorInteger[#][[-1, 1]] &,
TakeWhile[Range[2^n, 2^n - 20, -1], # > 0 &]], # > 0 &], {n, 36}] (* Michael De Vlieger, Jul 31 2017 *)

A102640 Compute the greatest prime factors (GPFs, A006530()) of j + 2^n for j = 0, 1, ..., L. a(n) is the maximal length L of such a sequence in which the greatest prime factors are increasing with increasing j.

Original entry on oeis.org

2, 2, 4, 2, 3, 2, 2, 2, 3, 2, 2, 4, 2, 3, 2, 2, 3, 4, 2, 3, 3, 6, 2, 3, 2, 4, 2, 2, 3, 4, 2, 3, 3, 2, 2, 4, 2, 4, 2, 2, 2, 4, 2, 3, 4, 4, 2, 4, 2, 3, 3, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 3, 4, 2, 3, 4, 4, 2, 4, 2, 3, 2, 2, 4, 4, 2, 3, 3, 4, 2, 2, 2, 3, 2, 4, 2, 3, 2, 2, 3, 2, 2, 2, 3, 4, 2, 4, 2, 3, 2, 2, 3

Offset: 1

Labos Elemer, Jan 21 2005

nonn

A006530(2^n)=2 is a local minimum. Going either upward or downward with the argument, the greatest prime factors are increasing for a while.

			For n = 12: 2^10 = 4096. The greatest prime factors of 4096, 4097, 4098, 4099 are as follows: {2, 241, 683, 4099}. A006530(4100) = 41 is already smaller than A006530(4099). Thus the length of increasing GPF sequence is 4 = a(12).

Cf. A006530, A102641, A102642, A102643, A102644.

With[{nn = 12}, Table[Function[k, 1 + LengthWhile[#, # > 0 &] &@ Differences@ Array[FactorInteger[#][[-1, 1]] &, nn, k]][2^n], {n, 105}]] (* Michael De Vlieger, Jul 24 2017 *)

A102641 Compute the greatest prime factors (GPFs, A006530()) of -j + 2^n for j = 0, 1, ..., L. a(n) is the maximal length L of such a sequence in which the greatest prime factors are increasing with decreasing j.

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 2, 3, 2, 4, 2, 4, 2, 4, 2, 2, 2, 3, 2, 4, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 3, 2, 4, 4, 4, 2, 3, 4, 4, 2, 3, 2, 4, 2, 2, 2, 4, 2, 3, 2, 4, 2, 4, 6, 2, 4, 2, 2, 3, 2, 3, 3, 2, 2, 3, 2, 3, 2, 3, 2, 4, 2, 3, 6, 4, 2, 3, 2, 3, 4, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 4, 2, 4, 4, 4, 2, 3, 3, 4, 2, 4, 2, 3, 5

Offset: 1

Labos Elemer, Jan 21 2005

nonn

A006530(2^n)=2 is a local minimum. Going either upward or downward with the argument, the largest prime factors are increasing for a while. Here the maximal length of increasing greatest-prime-factor sequences are given when going downward with the arguments. Compare with A102640.

			For n = 12: 2^10 = 4096. The greatest prime factors of 4096, 4095, 4094, 4093 are as follows: {2, 13, 89, 4093}. A006530(4092) = 31 is already smaller than A006530(4093). Thus the length of the increasing GPF sequence is 4 = a(12).

Cf. A006530, A102640, A102642, A102643, A102644.

With[{nn = 12}, Table[Function[k, 1 + LengthWhile[#, # > 0 &] &@ Differences@ Table[FactorInteger[m][[-1, 1]], {m, k, k - nn, -1}]][2^n], {n, 105}] ] (* Michael De Vlieger, Jul 24 2017 *)

A102642 a(n) = A102640(n) + A102641(n) - 1.

Original entry on oeis.org

2, 3, 5, 5, 4, 5, 3, 4, 4, 5, 3, 7, 3, 6, 3, 3, 4, 6, 3, 6, 4, 7, 3, 6, 3, 5, 3, 5, 4, 7, 3, 5, 4, 5, 5, 7, 3, 6, 5, 5, 3, 6, 3, 6, 5, 5, 3, 7, 3, 5, 4, 5, 3, 6, 7, 3, 6, 3, 3, 4, 3, 5, 4, 3, 4, 6, 3, 5, 5, 6, 3, 7, 3, 5, 7, 5, 5, 6, 3, 5, 6, 5, 3, 4, 3, 5, 3, 6, 3, 5, 3, 5, 4, 5, 5, 5, 4, 6, 4, 7, 3, 6, 3, 4, 7

Offset: 1

Labos Elemer, Jan 21 2005

nonn

A006530(2^n)=2 is a local minimum. Actual sequence displays the "width of valley" between the two nearest peaks of largest prime divisors. At the bottom of valley lies the number 2, the minimum.

			n=12: 2^10=4096. The greatest prime divisors of numbers around 4096 [both downward and upward] are as follows: {31, 4093, 89, 13, 2, 241, 683, 4099, 41}. The length of relevant sequence, i.e., between peaks 4093 and 4099 is 7, thus a(12)=7.

Cf. A006530, A102640, A102641, A102643, A102644.

With[{nn = 12, lim = 105}, Map[Total@ # - 1 &, Transpose@ {Table[Function[k, 1 + LengthWhile[#, # > 0 &] &@ Differences@ Array[FactorInteger[#][[-1, 1]] &, nn, k]][2^n], {n, lim}], Table[Function[k, 1 + LengthWhile[#, # > 0 &] &@ Differences@ Table[FactorInteger[m][[-1, 1]], {m, k, k - nn, -1}]][2^n], {n, lim}]}]] (* Michael De Vlieger, Jul 30 2017 *)

cp's OEIS Frontend

A102644 A006530(x)=2 is a local minimum if x=2^n. Running downward with argument x started at 2^n, the largest prime divisor should increase. The value of first peak is a(n).

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A102640 Compute the greatest prime factors (GPFs, A006530()) of j + 2^n for j = 0, 1, ..., L. a(n) is the maximal length L of such a sequence in which the greatest prime factors are increasing with increasing j.

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A102641 Compute the greatest prime factors (GPFs, A006530()) of -j + 2^n for j = 0, 1, ..., L. a(n) is the maximal length L of such a sequence in which the greatest prime factors are increasing with decreasing j.

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A102642 a(n) = A102640(n) + A102641(n) - 1.

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